1) In the case of one independent variable x, a differential is a form A(x)dx. If there is a function ##f(x)## such that its derivative is ##f'(x)=A(x)## then the differential is exact and it is written as ##df=f'(x)dx=A(x)dx##.

2) in the case of many variables lets say in the case of 3 independent variables, call them x,y,z a differential is a form ##A(x,y,z)dx+B(x,y,z)dy+C(x,y,z)dz##.

If there is a function ##f(x,y,z)## such that its corresponding partial derivatives with respect to x,y,z equal A,B,C then that differential is called an exact differential and is written a ##df##.

if there is not such a function f then the differential is called an inexact differential and can be written as ##\delta \vec {F}=\vec{F} \cdot d\vec{r}## where F is the vector in ##R^3## with ##\vec{F}=A(x,y,z)\vec{x}+B(x,y,z)\vec{y}+C(x,y,z)dz## and ##d\vec{r}=\vec{x}dx+\vec{y}dy+\vec{z}dz##

The ##\delta x## denotes rather a variation in variational calculus, e.g., in the Lagrange formalism of classical mechanics, where you have an action functional
##A[x]=\int_{t_1}^{t_2} \mathrm{d} t L(x,\dot{x}).##
Then ##\delta x## is a little distortion of a given path. You can define functional derivatives as derivatives of functionals rather in an analogous way as you define partial derivatives of multivariate functions. In the latter case you have independent variables ##x_j## with a discrete index ##j \in \{1,2,\ldots, n \}##, while in the former case you can take ##t## in ##x(t)## (defining a trajectory) as a kind of "continuous index".